Anomalies , Unitarity and Quantum Irreversibility

The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and 2R, with coefficients, properly normalized, called c, a and a ′ , the latter being ambiguously defined by an additive constant. Unitarity and positivity properties of the induced actions allow us to show that the total RG flows of a and a ′ are equal and therefore the a ′-ambiguity can be consistently removed through the identification a ′ = a. The picture that emerges clarifies several long-standing issues. The interplay between unitarity and renormalization implies that the flux of the renormalization group is irreversible. A monotonically decreasing a-function interpolating between the appropriate values is naturally provided by a ′. The total a-flow is expressed non-perturbatively as the invariant (i.e. scheme-independent) area of the graph of the beta function between the fixed points. We test this prediction to the fourth loop order in perturbation theory, in QCD with N f 11/2 N c and in supersymmetric QCD. There is agreement also in the absence of an interacting fixed point (QED and ϕ 4-theory). Our arguments do not seem to prove that a is strictly positive, but put a lower bound to its value.